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Journal of the London Mathematical Society 1999 60(2):321-332; doi:10.1112/S0024610799007875
© 1999 by London Mathematical Society
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© The London Mathematical Society

Lacunary Polynomials, Multiple Blocking Sets and Baer Subplanes

A. Blokhuis, L. Storme and T. Szonyi

Technical University Eindhoven PO Box 513, 5600 MB Eindhoven, Netherlands aartb{at}win.tue.nl
University of Gent, FCW Galglaan 2, 9000 Gent Belgium ls{at}cage.rug.ac.be
Department of Computer Science, Eötvös Loránd University Múzeum krt. 6–8, H-1088 Budapest, Hungary szonyi{at}cs.elte.hu

Received 21 March 1997. Revision received 28 January 1998.

New lower bounds are given for the size of a point set in a Desarguesian projective plane over a finite field that contains at least a prescribed number s of points on every line. These bounds are best possible when q is square and s is small compared with q. In this case the smallest set is shown to be the union of disjoint Baer subplanes. The results are based on new results on the structure of certain lacunary polynomials, which can be regarded as a generalization of Rédei's results in the case when the derivative of the polynomial vanishes.


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